On s-dimensional incomplete Kloosterman sums

In this paper we utilize the estimation of number of solutions of congruence to obtain the upper bound of incomplete Kloosterman sums, which improves the Shparlinski's result and removes the parameter r.     

A generalization of power moments of Kloosterman sums

We find an expression for a sum which can be viewed as a generalization of power moments of Kloosterman sums studied by Kloosterman and Salié. Received: 24 March 2006     

Using Stepanov's method for exponential sums involving rational functions

For any additive character ψ and multiplicative character χ on a finite field F_q, and rational functions f,g in F_q(x), we show that the elementary Stepanov-Schmidt method can be used to obtain the corresponding Weil bound for the sum ∑_x∈Fq?Sχ(g(x))ψ(f(x)) where S is the set of the poles of f and g. We also determine precisely the number of characteristic values ω_i of modulus q^1/2 and the number of modulus 1.     

On Weil's proof of the bound for Kloosterman sums

While most proofs of the Weil bound on one-variable Kloosterman sums over finite fields are carried out in all characteristics, the original proof of this bound, by Weil, assumes the characteristic is odd. We show how to make Weil's argument work in even characteristic, for both ordinary Kloosterman sums and sums twisted by a multiplicative character.     

Character sums with exponential functions over smooth numbers

We give nontrivial bounds in various ranges for exponential sums of the form

     

On certain exponential sums over primes

Let f(x) be a real valued polynomial in x of degree k?4 with leading coefficient α. In this paper, we prove a non-trivial upper bound for the quantity

     

On character sums and the exceptional set of a congruence problem

We prove an estimate of character sums. This bound and the method of solving multiplicative ternary problems are used to obtain new results about the cardinality of an exceptional set of a congruence problem modulo a prime p. In particular, we show that “almost all” residue classes modulo p are representable in the form , where      

Incomplete quadratic exponential sums in several variables

We consider incomplete exponential sums in several variables of the form

     

On character sums over flat numbers

Let q?2 be an integer, χ be any non-principal character mod q, and H=H(q)?q. In this paper the authors prove some estimates for character sums of the form

     

Kloosterman sums and traces of singular moduli

We give a new proof of some identities of Zagier relating traces of singular moduli to the coefficients of certain weakly holomorphic half integral weight modular forms. These identities play a central role in Zagier's work on the infinite product isomorphism introduced by Borcherds. In addition, we derive a simple expression for writing twisted traces of singular moduli as infinite series.     

Regularity of patterns in the factorization of n!

Consider the multiplicities ep₁(n),ep₂(n),…,epk(n) in which the primes p₁,p₂,…,pk appear in the factorization of n!. We show that these multiplicities are jointly uniformly distributed modulo (m₁,m₂,…,mk) for any fixed integers m₁,m₂,…,mk, thus improving a result of Luca and St?nic? [F. Luca, P. St?nic?, On the prime power factorization of n!, J. Number Theory 102 (2003) 298-305]. To prove the theorem, we obtain a result regarding the joint distribution of several completely q-additive functions, which seems to be of independent interest.     

On character sums of binary quadratic forms

We establish character sum bounds of the form

     

A generating function of higher-dimensional Apostol-Zagier sums and its reciprocity law

We introduce higher-dimensional Dedekind sums with a complex parameter z, generalizing Zagier's higher-dimensional Dedekind sums. The sums tend to Zagier's higher-dimensional Dedekind sums as z→∞. We show that the sums turn out to be generating functions of higher-dimensional Apostol-Zagier sums which are defined to be hybrids of Apostol's sums and Zagier's sums. We prove reciprocity law for the sums. The new reciprocity law includes reciprocity formulas for both Apostol and Zagier's sums as its special case. Furthermore, as its application we obtain relations between special values of Hurwitz zeta function and Bernoulli numbers, as well as new trigonometric identities.     

Directional wavelets on n-dimensional spheres

Directional Poisson wavelets, being directional derivatives of Poisson kernel, are introduced on n-dimensional spheres. It is shown that, slightly modified and together with another wavelet family, they are an admissible wavelet pair according to the definition derived from the theory of approximate identities. We investigate some of the properties of directional Poisson wavelets, such as recursive formulae for their Fourier coefficients or explicit representations as functions of spherical variables (for some of the wavelets). We derive also an explicit formula for their Euclidean limits.     

A note on character sums

We investigate certain character sums and prove some discrepancy-type inequalities for incomplete sums.      

On n-dimensional sampling theorems

Two n-dimensional nonuniform-sampling theorems are proved, which possess additional freedom for constructing recovery functions. One of these implies the n-dimensional uniform-sampling theorem for functions with band-limited Fourier transforms, which also has been proved here through distribution theory. Balth van der Pol's 1959 conjecture on nonuniform sampling for time function is a consequence of these two theorems. For either case, the recovery functions are analytic.     

q-Counting n-dimensional lattice paths

n-dimensional lattice paths are enumerated by generating functions which are Gaussian multinomial coefficients in the case of unrestricted paths. Convolutions for path counts are studied which yield a q-Vandermonde convolution and a determinant of Gaussian multinomial coefficients as the generating function for certain restricted paths.